1549380323-Statistical Mechanics Theory and Molecular Simulation

Time-dependent perturbation theory 535

between the Schr ̈odinger and Heisenberg pictures. Recall that inthe Schr ̈odinger pic-
ture, operators are static and the state evolves in time, while the opposite is true in the
Heisenberg picture. In the interaction picture, both the state vector and the operators
evolve in time.
The transformation of an operatorAˆfrom the Schr ̈odinger picture to the interac-
tion picture is given by

AˆI(t) = eiHˆ^0 (t−t^0 )/ ̄hAˆe−iHˆ^0 (t−t^0 )/ ̄h, (14.2.3)

which is equivalent to an equation of motion of the form

dAˆI(t)

dt

=

1

i ̄h

[AˆI(t),Hˆ 0 ]. (14.2.4)

Note the similarity between this transformation and that of eqn. (9.2.58) between
the Schr ̈odinger and Heisenberg pictures. Eqn. (14.2.4) indicatesthat the time evolu-
tion of operators in the interaction picture is determined solely by the unperturbed
HamiltonianHˆ 0.
Eqn. (14.2.2) represents a transformation of the state vector between the Schr ̈odinger
and interaction pictures. The time-evolution equation for the state|Φ(t)〉can be de-
rived by substituting eqn. (14.2.2) into eqn. (14.2.1), which yields

(

Hˆ 0 +Hˆ 1 (t)

)

e−i

Hˆ 0 (t−t 0 )/ ̄h

|Φ(t)〉=i ̄h

∂

∂t

e−i

Hˆ 0 (t−t 0 )/ ̄h

|Φ(t)〉

(

Hˆ 0 +Hˆ 1 (t)

)

e−i

Hˆ 0 (t−t 0 )/ ̄h

|Φ(t)〉=Hˆ 0 e−i

Hˆ 0 (t−t 0 )/ ̄h

|Φ(t)〉+ e−i

Hˆ 0 (t−t 0 )/ ̄h

i ̄h

∂

∂t

|Φ(t)〉

Hˆ 1 (t)e−iHˆ^0 (t−t^0 )/ ̄h|Φ(t)〉=e−iHˆ^0 (t−t^0 )/ ̄hi ̄h∂

∂t

|Φ(t)〉. (14.2.5)

Multiplying on the left byeiHˆ^0 (t−t^0 )/ ̄hyields

ei

Hˆ 0 (t−t 0 )/ ̄hˆ

H 1 (t)e−i

Hˆ 0 (t−t 0 )/h ̄

|Φ(t)〉=i ̄h

∂

∂t

|Φ(t)〉. (14.2.6)

According to eqn. (14.2.3), the exp[iHˆ 0 (t−t 0 )/ ̄h]Hˆ 1 (t) exp[−iHˆ 0 (t−t 0 )/ ̄h] is the
interaction-picture representation of the perturbation Hamiltonian, which we will de-
note asHˆI(t). The time evolution of the state vector in the interaction picture is,
therefore, given by a Schr ̈odinger equation of the form

HˆI(t)|Φ(t)〉=i ̄h∂

∂t

|Φ(t)〉. (14.2.7)

Eqn. (14.2.7) shows that this time evolution is determined entirely by the interaction-
picture representation of the perturbation,HˆI(t). According to eqn. (14.2.2), the initial
condition to eqn. (14.2.7) is|Φ(t 0 )〉=|Ψ(t 0 )〉. In the next subsection, we will develop
an iterative solution to eqn. (14.2.7), which will reveal the detailed structure of the
propagator for time-dependent systems.